Problem: Given $ m \angle RPS = 7x + 23$, $ m \angle QPR = 2x + 10$, and $ m \angle QPS = 168$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {2x + 10} + {7x + 23} = {168}$ Combine like terms: $ 9x + 33 = 168$ Subtract $33$ from both sides: $ 9x = 135$ Divide both sides by $9$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 7({15}) + 23$ Simplify: $ {m\angle RPS = 105 + 23}$ So ${m\angle RPS = 128}$.